Lorenzo Valla's rhetorical reform of logic resulted in important changes in sixteenth-century mathematical sciences, and not only in mathematical education and in the use of mathematics in other sciences, but also in mathematical theory itself. Logic came to be identified with dialectic, syllogisms with enthymemes and necessary truth with the limit case of probable truth. Two main ancient authorities mediated between logical and mathematical concerns: Cicero and Proclus. Cicero's 'common notions' were identified with Euclid's axioms, so that mathematics could be (...) viewed as core knowledge shared by all humankind. Proclus' interpretation of Euclid's axioms gave rise to the idea of a universal human natural light of reasoning and of a mathesis universalis as a basic mathematics common to both arithmetic and geometry and as an art of thinking interpretable as algebra. (shrink)

The admissibility of Ackermann's rule γ is one of the most important problems in relevant logics. The admissibility of γ was first proved by an algebraic method. However, the development of Routley-Meyer semantics and metavaluational techniques makes it possible to prove the admissibility of γ using the method of normal models or the method using metavaluations, and the use of such methods is preferred. This paper discusses an algebraic proof of the admissibility of γ in relevant modal logics based on (...) modern algebraic models. (shrink)

Famous for the number-theoretic first-order statement known as Goodstein's theorem, author R. L. Goodstein was also well known as a distinguished educator. With this text, he offers an elementary treatment that employs Boolean algebra as a simple medium for introducing important concepts of modern algebra. The text begins with an informal introduction to the algebra of classes, exploring union, intersection, and complementation; the commutative, associative, and distributive laws; difference and symmetric difference; and Venn diagrams. Professor Goodstein proceeds (...) to a detailed examination of three different axiomatizations, and an outline of a fourth system of axioms appears in the examples. The final chapter, on lattices, examines Boolean algebra in the setting of the theory of partial order. Numerous examples appear at the end of each chapter, with full solutions at the end. (shrink)

This is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel’s incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. (...) We relate the algebraic notion of neat embeddings (a notion special to cylindric algebras) to the metalogical ones of provability, interpolation and omitting types in variants of first logic. Another novelty that occurs here is relating the algebraic notion of atom-canonicity for a class of boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. A hitherto unpublished application of algebraic logic to omitting types of first order logic is given. Proofs are included when they serve to illustrate certain concepts. Several open problems are posed. We have tried as much as possible to avoid exploring territory already explored in the survey articles of Monk [93] and Németi [97] in the subject. (shrink)

According to the algebraic approach to spacetime, a thoroughgoing dynamicism, physical fields exist without an underlying manifold. This view is usually implemented by postulating an algebraic structure (e.g., commutative ring) of scalar-valued functions, which can be interpreted as representing a scalar field, and deriving other structures from it. In this work, we point out that this leads to the unjustified primacy of an undetermined scalar field. Instead, we propose to consider algebraic structures in which all (and only) physical fields are (...) primitive. We explain how the theory of natural operations in differential geometry—the modern formalism behind classifying diffeomorphism-invariant constructions—can be used to obtain concrete implementations of this idea for any given collection of fields. For concrete examples, we illustrate how our approach applies to a number of particular physical fields, including electrodynamics coupled to a Weyl spinor. (shrink)

The article offers an analysis of Simone Weil's philosophy of mathematics. Weil's reflection starts from a critique of Bourbaki's programme, led by her brother André: the "mechanical attention" Bourbaki considered an advantage of their treatment of mathematics was for her responsible for the incomprehensibility of modern algebra, and even a cause of alien-ation and social oppression. On the contrary, she developed her pivotal concept of 'atten-tion' with the aim of approaching mathematical problems in order to make "progress in another (...) more mysterious dimension". In the Pythagorean 'crisis of incommensurables', Weil saw the possibility of defining the relationships between things in terms that are not exclusively numerical. This implies drawing an analogy between mathematical relation-ships and God's relationship with mankind (logos), the basis of a 'supernatural' reformu-lation of the entire scientific understanding of the world. The consequence is a critique of machinism and the possibility to contrast algorithmic reason with a "supernatural reason". (shrink)

The study that George Lakoff and Rafael Núñez call "idea analysis" and begin in their recent book Where mathematics comes from is intended to dissect mathematical concepts into their metaphorical parts, where metaphor is used in the cognitive-science sense promoted by Lakoff and Mark Johnson in Metaphors we live by and subsequent works by each of them and together. Lakoff and Núñez's analysis of the (modern) algebraic concept of group is based on the attribution to contemporary mathematics of what will (...) be widely recognizable by their name for it, the folk theory of essences. I argue that this philosophical basis for their analysis is spurious and supply an alternative analysis of the same concept within their "metaphorical" paradigm but without essences. This analysis, which I hope is more viable than theirs, is intended to support the general applicability of the paradigm by freeing it from outmoded philosophical baggage. (shrink)

“Ordinary algebra in its modern developments,” Whitehead observed in 1897, “is studied as being a large body of propositions, inter-related by deductive reasoning, and based upon conventional definitions which are generalizations of fundamental conceptions.” The use of ‘based upon’ here is perhaps too weak, for some “propositions” must of course be picked out as determinative of the kind of algebra in question by way of axioms. The definitions are then ancillary devices of notational abbreviation and may or may (...) not be of terms or phrases previously introduced, and may or may not be of a “generalized” form. “Thus a science is being created [ordinary algebra in its modern developments],” Whitehead continues, “which by reason of its fundamental character has relation to almost every event, phenomenal or intellectual, which can occur…. Such algebras are mathematical sciences which are not essentially concerned with number or quantity; and this bold extension beyond the traditional domain of pure quantity forms their peculiar interest. The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thought, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.”. (shrink)

Summary This paper investigates the fate of Thomas Harriot's algebra after his death in 1621 and, in particular, the largely unsuccessful efforts of seventeenth-century mathematicians to promote it. The little known surviving manuscripts of Nathaniel Torporley have been used to elucidate the roles of Torporley and Walter Warner in the preparation of the Praxis, and a partial translation of Torporley's important critique of the Praxis is offered here for the first time. The known whereabouts of Harriot's mathematical papers, both (...) originals and copies, during the seventeenth century and later are summarised. John Wallis's controversial 1685 account of Harriot's algebra is examined in detail and it is argued that John Pell's influence on Wallis was far more significant than has previously been realised. The paper ends with a reassessment of Harriot's underrated and important contribution to the development of modern algebra. (shrink)

Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first‐order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research on (...) increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WA's with set‐theoretic projection elements – where this is not the case. These “logical” connectives are set‐theoretic counterparts of the axiomatic quasi‐projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi‐equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non‐recursively enumerable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

The geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that the geometrical algebra interpretation should be reinstated as a viable historical hypothesis.

This article reconstructs the theory of the Soviet-American psychologist Vladimir Lefebvre as part of the neocybernetic movement. In particular, I propose to explore such elements of his research of the 1970s—1990s as systemic vision; reflexive analysis; a search for holistic configuration and Janus cosmology; and the realization of neocybernetics. An interest in the reflexive structures of cognition and action led Lefebvre to an understanding of the limited nature of the world’s scientific picture. The conflicting objects he studied proved too complex (...) for behavioral modeling and forecasting. Lefebvre therefore expanded the boundaries of the application of his formal theory of intelligence, relying on functionalist positions, which led him to the science-fiction narrative of “The Big Correction.” I contend that Lefebvre’s works not only propose an original Soviet version of the neocybernetics the 1970s but develop their own model of inhumanism long before it appeared on the scene of modern philosophy. Lefebvre’s formal theory, reinforced by thermodynamics and elements of Russian cosmism, builds cognitive behavior into the teleology of the Big Correction, which involves the idea of overcoming the heat death of the universe. The activity of cosmic intelligence, which Lefebvre equates with magnetic-plasma structures in the finished areas of the universe, is aimed at cooperation and prolonging the existence of life and the intellect. (shrink)

In this paper, we examine the relationship between general relativity and the theory of Einstein algebras. We show that according to a formal criterion for theoretical equivalence recently proposed by Halvorson and Weatherall, the two are equivalent theories.

Despite the efforts of many individuals, the disciplines of modern physics and number theory have remained largely divorced, in the sense that the experimentally verified theories of quantum physics and gravity are written in the language of linear algebra and advanced calculus, without reference to several established branches of pure mathematics. This absence raises questions as to whether or not pure mathematics has undiscovered application to physical modeling that could have far reaching implications for human scientific understanding. In this (...) paper, we review physical interpretations of number theoretic concepts developed in the twentieth century, in an attempt to help bridge the divide between pure mathematical truth and testable physical theory. Specifically, the relevance of L-functions and modular forms to the physics of quantum and classical physical systems is addressed, with the objective of motivating general interest among physicists in these mathematical concepts. (shrink)

Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived algebraically. The adjective “geometric” likely entailed many things for Scott, from her careful illustration of diagrams to her choice of references (...) and citations. This article will focus on Scott’s striking and consistent use of geometric to describe a reality of dynamic points, lines, planes, and spaces that could be manipulated analogously to physical objects. By providing geometric interpretations of algebraic derivations, Scott committed to an early-nineteenth-century aesthetic vision of a “whole” analytical geometry that she adapted to modern research areas. (shrink)

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

History is familiar with great scientific traditions which have been substantial, effective, cumulative and progressive.* At the level of great eras of civilization, extensive and not episodic phenomena, very ancient Chinese science, Greek science and Arab science are objects of investigation for historical erudition, but also for the scientific historian and the philosopher of sciences. Many of the elements of these systems were the source of “modern science”, as it is called, or are integral parts ol’ this system of knowledge (...) and practice which has been constructed over a little more than three centuries. Diachronically, the “scientific revolution” of the classical age in Europe does not signify a total break with what preceded it: Chinese astronomy, Arab algebra and the hospital organization of the Islamic world were used, even if the ideology of scientists when they wrote their own history, particularly from the 19th century onward, tends to make of modern science a purely European phenomenon. (shrink)

Dans cet article, j’étudie la façon dont Pierre de La Ramée aborde la question de l’analyse mathématique, d’abord dans ses écrits de logicien, puis dans ses traités mathématiques. Dans les Scholae mathematicae (1569), en reprenant un argumentaire qui lui avait servi dans les controverses des années 1550 concernant la « méthode unique», il est conduit à la conclusion que l’analyse n’a aucune valeur démonstrative. Ses recherches sur les significations diverses du mot analysis et sur la nature de l’analyse dans les (...) mathématiques grecques m’apparaissent pourtant comme une condition préalable pour que soit reconnue au xviie siècle la fécondité de la méthode analytique en géométrie. (shrink)

The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception (...) of rigor and intuition. Unlike what is often assumed today, from their perspective, rigor is neither opposed to intuition nor understood as a unitary phenomenon –Enriques distinguishes between _ small-scale rigor _ and _ large-scale rigor _ and Severi between _ formal rigor _ and _ substantial rigor _. Finally, we turn to the notion of mathematical objectivity. We draw from our case study in order to advance a multi-dimensional analysis of objectivity. Specifically, we s... (shrink)

The modern use of algebra to describe geometric ideas is discussed with particular reference to the constructions of Grassmann and Hamilton and the subsequent algebras due to Clifford. An Einstein addition law for nonparallel boosts is shown to follow naturally from the use of the representation-independent form of the geometric algebra of space-time.

This paper deals with the algebraization of multi-signature equational logic in the context of the modern theory of categorical abstract algebraic logic. Two are the novelties compared to traditional treatments: First, interpretations between different algebraic types are handled in the object language rather than the metalanguage. Second, rather than constructing the type of the algebraizing class of algebras explicitly in an ad-hoc universal algebraic way, the whole clone is naturally constructed using categorical algebraic techniques.

This paper augments Hailperin's substantial efforts to place Boole's algebra of logic on a solid footing. Namely Horn sentences are used to give a modern formulation of the principle that Boole adopted in 1854 as the foundation for his algebra of logic—we call this principle The Rule of 0 and 1.

Ronny Desmet & Michel Weber (edited by), Whitehead. The Algebra of Metaphysics. Applied Process Metaphysics Summer Institute Memorandum, Louvain-la-Neuve, Les Éditions Chromatika, 2010. (978-2-930517-08-7 ; 378 p. ; 40 € ; ) Drawing upon the major Harvard works —Science and the Modern World (1925), Process and Reality (1929) and Adventures of Ideas (1933)—, the essays gathered here on the occasion of the creation of the Applied Process Metaphysics Summer Institute, seek, first, to introduce into Whitehead’s thought by clarifying what (...) is at stake in his philosophy and by providing a synoptic vision of his key categories in light of their historical development and, second, to foster a creative dialogue among all participants. These essays give the opportunity to travel through most aspects of Whitehead’s legacy: anthropology, ecology, education, epistemology, metaphysics, psychology, political theory and relativity physics. Ronny Desmet holds an M.A. in mathematics (1983), an M.A. and a Ph.D. in philosophy (2005 and 2010). He is currently research assistant at the Centre of Logic and Philosophy of Science, Free University of Brussels. His research and publications have focused on Whitehead’s philosophy of mathematics and relativity. Michel Weber holds an M.A. and a Ph.D. in philosophy (1991 and 1997). He is the director of the Centre for Philosophical Practice “Chromatiques whiteheadiennes.” He has published more than 50 books, e.g., Whitehead’s Pancreativism. The Basics (2006) and the Handbook of Whiteheadian Process Thought (2008). (shrink)

I argue against the currently prevalent view that algebraic quantum field theory (AQFT) is the correct framework for philosophy of quantum field theory and that “conventional” quantum field theory (CQFT), of the sort used in mainstream particle physics, is not suitable for foundational study. In doing so, I defend that position that AQFT and CQFT should be understood as rival programs to resolve the mathematical and physical pathologies of renormalization theory, and that CQFT has succeeded in this task and AQFT (...) has failed. I also defend CQFT from recent criticisms made by Doreen Fraser. (shrink)

The first beginnings of modern logic in Croatia are recognizable as early as in the middle of the 19th century in Vatroslav Bertić. At the turn of the 20th century, Albin Nagy, who was teaching in Italy, made contributions to algebraic logic and to the philosophy of logic. At that time, a distinctive author Mate Meršić stood out, also working on algebraic logic. In the Croatian academic philosophy, until the publication of Gajo Petrović's textbook (1964) and the contributions by Heda (...) Festini, a critical or indifferent attitude toward modern logic prevailed. The first comprehensive modern logic textbook in Croatian was written by mathematician Vladimir Devidé (1964), known, for example, for his axiomatizations of natural numbers and classical propositional logic. In the Croatian philosophy of the 1980s and 1990s, influential logicians are Goran Švob and mathematician Zvonimir Šikić. At the turn of the 21st century, logical investigations are being intensified and extended to various branches of the contemporary logic. (shrink)

We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and (...) set theory, which undoubtedly point towards the mathematics of twentieth and twenty-first centuries. (shrink)

The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of (...) rigor and intuition. Unlike what is often assumed today, from their perspective, rigor is neither opposed to intuition nor understood as a unitary phenomenon – Enriques distinguishes between small-scale rigor and large-scale rigor and Severi between formal rigor and substantial rigor. Finally, we turn to the notion of mathematical objectivity. We draw from our case study in order to advance a multi-dimensional analysis of objectivity. Specifically, we suggest that various types of rigor may be associated with different conceptions of objectivity: namely, objectivity as faithfulness to facts and objectivity as intersubjectivity. (shrink)

ABSTRACT In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would suggest (...) arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (shrink)

The CPT theorem states that any causal, Lorentz-invariant, thermodynamically well-behaved quantum field theory must also be invariant under a reflection symmetry that reverses the direction of time, flips spatial parity, and conjugates charge. Although its physical basis remains obscure, CPT symmetry appears to be necessary in order to unify quantum mechanics with relativity. This paper attempts to decipher the physical reasoning behind proofs of the CPT theorem in algebraic quantum field theory. Ultimately, CPT symmetry is linked to a systematic reversal (...) of the C*-algebraic Lie product that encodes the generating relationship between observables and symmetries. In any physically reasonable relativistic quantum field theory it is always possible to systematically reverse this generating relationship while preserving the dynamics, spectra, and localization properties of physical systems. Rather than the product of three separate reflections, CPT symmetry is revealed to be a single global reflection of the theory’s state space. (shrink)

In this paper, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary differential equations. This leads us to the work of Mihailo Petrović on algebraic differential equations and in particular the geometric ideas expressed in his polygon method from the final years of the nineteenth century, which have been left completely unnoticed by the experts. This concept, also developed independently and in a somewhat different direction by Henry Fine, generalizes the (...) famous Newton–Puiseux polygonal method and applies to algebraic ODEs rather than algebraic equations. Although remarkable, the Petrović legacy has been practically neglected in the modern literature, although the situation is less severe in the case of results of Fine. Therefore, we study the development of the ideas of Petrović and Fine and their places in contemporary mathematics. (shrink)

Françios Viète was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek (...) geometric equalities. His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviews 91 THE ROOTS OF MODERN LOGIC ALASDAIR URQUHART Philosophy/ U. ofToronto Toronro, ON, Canada M5S IAI [email protected] I. Grattan-Guinness. The Searchfor Mathematical Roots,r870--r940: logics, Set Theoriesand the Foundations of Mathematicsfrom Cantor through Russellto Godel Princeron: Princeton U. P.,2000. Pp. xiv,690. us$45.oo. Grattan-Guinness's new hisrory of logic is a welcome addition to the literature. The title does not quite do justice ro the book, since it begins with the (...) prehistory of English work in algebraic logic, including the work of the French analycicalschool, and extends to just after Godel's great incompleteness paper of 1931.The core of the book, though, is the philosophical and mathematical developmems leading up to and immediately following from chework ofWhicehead and Russell.Russell is ac chehearcof che book, and as a whole che hisrory forms an imporranr conrribucion ro Russell scudies. Commemarors on Russell have often confined themselves to a rather narrow hisrorical perspeccivein which Russell is seen as the (problemacic) heir of Gottlob Frege, and few ocher hisrorical figures (ocher than Peano) emer imo che picture. Graccan-Guinness corrects chis hisrorica.limbalance by placing Russell in che mucl1 wider context of the development of machemacics on che continent, and in parricu.laremphasizing strongly che key influence of Camor on 92 Reviews Russell's work. Cantor has usually received short shrift from philosophers, as unlike Frege,he appears rarher naive from rhe philosophical point of view. The story proper begins in Chaprer 2 wirh L-igrange'sversion of analysis in which rhe basic concepts were to be defined in rerms of algebraic manipularion of power series. This lead to rhe founding of rhe Analyrical Socierywhere Babbage, Herschel and Peacock were active. It is in chis English tradicion of algebraic analysis char rhe pioneering work of De Morgan and Boole found irs roors. Grarran-Guinness givesa derailed account of the work of both logicians, although his discussion of Boole's merhods does not seem entirely adequate. The question is:what are we to make of Boole'spuzzling insistence chat expressions like x + y are "uninterpretable", while he nevertheless manipulated them freely in his mathematical derivations? It is not correct to say that the addition sign can only link disjoint class symbols, since ir is belied by Boole's formal pracrice. A possible solution has been suggested by Hailperin, in his book on Boole's logic, in which he proposes interpreting the "uninterpretable" expressions as denoting signed mulrisets. Oddly, Grattan-Guinness refers to Hailperin 's work,' bur elsewhere (p. 42) adopts rhe view that Boole's addition sign could only link disjoint classes.The chapter concludes with brief accounts of the work of Cauchy, Weierstrass and Bolz.'lno. The next chapter is a detailed account of rhe work of Cantor and his creation of Mengenlehre. The origins of set theory in the theory of trigonometrical series, and rhe ensuing discoveryof rransfinire ordinal and cardinal numbers, are described clearly and succinctly. In addition, the chapter contains an account of Dedekind's philosophy of arithmetic and Cantor's philosophy of mathematics. Cantor's philosophy is an uneasy blend of formalism, platonism and idealism, and has tmderstandably aroused little enthusiasm among philosophers of matl1ematics, alrhough Michael Hallerr has recently smdied it in detail. GrammGuinness emphasizes, rhough, Cantor's magnificent marhematica.l achievements, in defining and clarifying basic conceprs such as measure, dimension and cardinality of sets. Chaprer 4 is a rather miscellaneous chapter, in which six partly independent, partly interrwined stories are told. It begins with developments in sectheory in Germany and France up to the turn of rhe century, goes on to discuss American logic in rhe work of C. S. Peirce and his students, and continues the rheme of algebraic logic with Schroder and his logic of relatives. The remainder of the chapter is given over to Frege, Husserl and Hilbert. The section on Frege is one of rhe more idiosyncratic parts of the book. Grattan-Guinness distinguishes berween Frege, "a mathematician who wrore in ' T Hailperin, Book's logic and Probability, 2nd ed. (Amsterdam: North-Holland, 1986). Reviews 93 German, in a markedly Platonic spirit", and Frege, "a philosopher of language and founder of... (shrink)

Although the notion of the crisis of European sciences has a general meaning, Husserl mainly focuses on this phenomenon in relation to the modern establishment of a mathematical natural science. However, he does not provide a definitive clarification of how its new method is specifically involved in bringing about such a crisis. Without trying to offer a faithful exegetical contribution, this paper further elaborates on Husserl’s analyses in the Krisis to give a possible answer to this question. After defining the (...) concept of crisis in general, I single out why algebraic thinking is the core of the method that essentially characterizes mathematical physics. Then, I compare this method to a case of pre-modern discipline (namely, Euclidean geometry) to show that the sedimentation of meaning at work in the progressive technization of algebra is exceptional and sufficient to explain the specificity of the modern character of the crisis of the exact sciences. The exclusively technical development of formal mathematics, in fact, elicits those shifts of meaning (such as objectivism) that phenomenology is required to amend to overcome the crisis and to establish an authentic form of knowledge. On the basis of these results, I conclude by suggesting a few consequences that are useful for the understanding of the project of phenomenology of the later Husserl. (shrink)

Bell inequalities, understood as constraints between classical conditional probabilities, can be derived from a set of assumptions representing a common causal explanation of classical correlations. A similar derivation, however, is not known for Bell inequalities in algebraic quantum field theories establishing constraints for the expectation of specific linear combinations of projections in a quantum state. In the paper we address the question as to whether a ‘common causal justification’ of these non-classical Bell inequalities is possible. We will show that although (...) the classical notion of common causal explanation can readily be generalized for the non-classical case, the Bell inequalities used in quantum theories cannot be derived from these non-classical common causes. Just the opposite is true: for a set of correlations there can be given a non-classical common causal explanation even if they violate the Bell inequalities. This shows that the range of common causal explanations in the non-classical case is wider than that restricted by the Bell inequalities. (shrink)